A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r) / sum_{s=0,...,q} ((-1)^s*binomial(2*q-s,s)*x^s), where q=1,2,....

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 13, 18, 1, 6, 9, 19, 38, 47, 1, 7, 11, 25, 58, 117, 123, 1, 8, 13, 31, 78, 187, 370, 322, 1, 9, 15, 37, 98, 257, 622, 1186, 843, 1, 10, 17, 43, 118, 327, 874, 2110, 3827, 2207, 1, 11, 19, 49, 138, 397, 1126, 3034, 7252, 12389, 5778, 1

Offset

0,2

Comments

Row indices q begin with 1, column indices n begin with 0.

Links

Table of n, a(n) for n=0..65.

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Formula

Conjecture. The n-th entry in row q is given by R_q(n) = 2^(2*n)*(sum_{j=1,...,n+1} (cos(j*Pi/(2*q+1)))^(2*n)), q >= 1, n >= 0.

Conjecture. G.f. for column n is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x, n >= 0.

Conjecture. 2*A185095(q,n) = A198632(2*q,n), q >= 1, n >= 0. - L. Edson Jeffery, Nov 23 2013

Example

Array begins as
1,  1,  1,  1,   1,    1, ...
2,  3,  7, 18,  47,  123, ...
3,  5, 13, 38, 117,  370, ...
4,  7, 19, 58, 187,  622, ...
5,  9, 25, 78, 257,  874, ...
6, 11, 31, 98, 327, 1126, ...
...

Crossrefs

Conjecture. Transpose of array A186740.

Conjecture. Rows 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).

Conjecture. Columns 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.

Cf. A209235.

Sequence in context: A125175 A210552 A193376 * A177888 A073020 A090349

Adjacent sequences: A185092 A185093 A185094 * A185096 A185097 A185098

Keyword

nonn,tabl

Author

L. Edson Jeffery, Jan 23 2012

Status

approved